91Ó°ÊÓ

Determine the following indefinite integrals. Check your work by differentiation. $$\int(5 s+3)^{2} d s$$

Find all functions \(f\) whose derivative is \(f^{\prime}(x)=x+1\).

Locating critical points a. Find the critical points of the following functions on the domain or on the given interval. b. Use a graphing utility to determine whether each critical point corresponds to a local maximum, local minimum, or neither. $$f(x)=\frac{4 x^{5}}{5}-3 x^{3}+5 \text { on }[-2,2]$$

Determine the following indefinite integrals. Check your work by differentiation. $$\int 5 m\left(12 m^{3}-10 m\right) d m$$

Evaluate the following limits. $$\lim _{x \rightarrow 0} \frac{\sin x-x}{7 x^{3}}$$

Find the intervals on which \(f\) is increasing and decreasing. $$f(x)=\cos ^{2} x \text { on }[-\pi, \pi]$$

Imagine a flat-bottomed cylindrical pot with a circular cross section of radius 4. A marble with radius \(0

Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error. $$1 / \sqrt{119}$$

An important question about many functions concerns the existence and location of fixed points. A fixed point of \(f\) is a value of \(x\) that satisfies the equation \(f(x)=x ;\) it corresponds to a point at which the graph of intersects the line \(y=x\). Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations. $$f(x)=5-x^{2}$$

Make a complete graph of the following functions. If an interval is not specified, graph the function on its domain. Use a graphing utility to check your work. $$f(x)=x \sqrt{x+4}$$